Finite Group Databasebeta

← #24630

Subgroups — #24630

Order 256.

453 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 453 conjugacy classes of subgroups — too many to draw. Only its 139 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#461C211yesmaximal, characteristic
2C2×C4×SD16C211yesmaximal, characteristic
3#2546C211yesmaximal, characteristic, complemented
4#2354C211yesmaximal, characteristic
5#2055C212yesmaximal
6#2013C212yesmaximal
7#1981C212yesmaximal, complemented
8#1956C211yesmaximal, characteristic, complemented
9#1939C211yesmaximal, characteristic, complemented
10#1927C212yesmaximal, complemented
11#1803C211yesmaximal, characteristic, complemented
12C22×SD16C2211yescharacteristic
13C22×SD16C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4⋊Q8C2212yes—
16C2×C4×Q8C2211yescharacteristic
17C2×C4×D4C2211yescharacteristic
18C2×#117—21no—
19#461—24no—
20C4×SD16—24no—
21C2×#106C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×C4⋊C8C2212yes—
24C2×#103C2211yescharacteristic
25C2×#103C2212yes—
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×#102C2211yescharacteristic
30C2×C4×C8C2211yescharacteristic
31#398—22no—
32#391C2212yes—
33#390C2212yes—
34#389—21no—
35#389—21no—
36#388—22no—
37#384C2211yescharacteristic
38#380—21no—
39C4×C4⋊C4C2212yes—
40C4×C4⋊C4C2211yescharacteristic
41#340C2212yes—
42#336C2212yes—
43C22×Q8C2312yes—
44C22×Q8C2311yescharacteristic
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C2×SD16—22no—
48C2×SD16—22no—
49C2×SD16—24no—
50C2×SD16—22no—
51C2×SD16—42no—
52C2×SD16—42no—
53C22×C8C2312yes—
54C22×C8C2311yescharacteristic
55C22×C8C2311yescharacteristic
56C4⋊Q8—24no—
57C4⋊Q8—22no—
58C4⋊Q8—22no—
59C4⋊Q8—42no—
60C4×Q8—22no—
61C4×Q8—22no—
62C4×D4—22no—
63C4×D4D414yes—
64#117—44no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4D411yescharacteristic
73C2×C4⋊C4D411yescharacteristic
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2312yes—
76C2×C4⋊C4C2311yescharacteristic
77C2×C22⋊C4—21no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C42—22no—
81C2×C42—21no—
82C2×C42C2312yes—
83C2×C42C2311yescharacteristic
84#106—22no—
85C4⋊C8—22no—
86C4⋊C8—42no—
87#103—24no—
88#103—22no—
89#103—24no—
90#103—42no—
91#103—42no—
92#102—22no—
93C4×C8—22no—
94#95—22no—
95#95—22no—
96#95—22no—
97#95D412yes—
98C24—21no—
99C2×Q8—21no—
100C2×Q8—21no—
101C2×Q8—22no—
102C2×Q8—21no—
103C2×Q8—21no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—21no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8#4112yes—
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—21no—
116C2×D4C2×D412yes—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—24no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4C2×D412yes—
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D412yes—
128C22×C4C2×D411yescharacteristic
129C22×C4C2411yescharacteristic
130SD16—28no—
131SD16—48no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—22no—
136C2×C8—21no—
137C2×C8—22no—
138C2×C8—21no—
139C2×C8—22no—
140C2×C8#4112yes—
141C4⋊C4—22no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—22no—
145C4⋊C4—21no—
146C4⋊C4—22no—
147C4⋊C4—44no—
148C4⋊C4—44no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—81no—
154C4⋊C4SD1614yescomplemented
155C4⋊C4C2×D412yes—
156C22⋊C4—24no—
157C22⋊C4—44no—
158C22⋊C4—82no—
159C42—22no—
160C42—22no—
161C42—42no—
162C42—44no—
163C42C2×D412yes—
164C42#4112yes—
165C23—22no—
166C23—41no—
167C23—41no—
168C23—41no—
169C23—41no—
170C23—41no—
171C23—41no—
172C23C22×D411yescharacteristic
173Q8—22no—
174Q8—22no—
175Q8—41no—
176Q8—41no—
177Q8—42no—
178Q8—42no—
179Q8—42no—
180Q8—42no—
181D4—22no—
182D4—22no—
183D4—41no—
184D4—41no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—48no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4#12112yes—
209C2×C4#12112yes—
210C2×C4C2×SD1614yes—
211C2×C4C2×SD1612yes—
212C2×C4#13712yes—
213C2×C4C22×D411yescharacteristic
214C2×C4C2×#4111yescharacteristic
215C2×C4C2×#4111yescharacteristic
216C2×C4C2×#4111yescharacteristic
217C2×C4H(5,𝔽2)11yescharacteristic
218C2×C4H(5,𝔽2)11yescharacteristic
219C8—22no—
220C8—41no—
221C8—42no—
222C22—42no—
223C22—42no—
224C22—42no—
225C22—81no—
226C22—81no—
227C22—81no—
228C22—81no—
229C22C2×#12111yescharacteristic
230C22#54611yescharacteristic
231C22#55011yescharacteristic
232C22C22×SD1611yescharacteristic
233C22C2×#13711yescharacteristic
234C22#57711yescharacteristic
235C22#57811yescharacteristic
236C4—24no—
237C4—22no—
238C4—42no—
239C4—42no—
240C4—42no—
241C4—82no—
242C4—82no—
243C4—81no—
244C4#46114yes—
245C4#52511yescharacteristic
246C4#53511yescharacteristic
247C4#55311yescharacteristic
248C4#55511yescharacteristic
249C2—82no—
250C2#241411yescharacteristic
251C2C2×#46111yescharacteristic
252C2#304611yescharacteristic
253C2#329811yescharacteristic
254C2#332911yescharacteristic
255C2#339211yescharacteristic
256C2#339811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #2546C2semidirect
7. #1981C2semidirect
8. #1956C2semidirect
9. #1939C2semidirect
10. #1927C2semidirect
11. #1803C2semidirect
154. C4⋊C4SD16semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#4612
C2×C4×SD162
#25462
#23542
#20552
#20132
#19812
#19562
#19392
#19272
#18032